Excited Vibrations of an Atmosphere. 151 
Following Lord liavleigh, all amplitudes concerned are treated 
as small, the mean temperature is taken as uniform over the 
whole of the spherical sheet considered, and of course the 
limitations imposed by the thinness of the sheet preclude any 
systematic motion of the air corresponding to trade-winds or 
the like. 
It will he assumed that in the free vibrations of the spherical 
sheet, the relation between density and pressure follows the 
adiabatic law : so that in the case of thermally excited 
vibrations the only transfer of heat of any account is that 
arising from the heatino-.effect of the sun and from radiation 
into space, no appreciable effect being produced by the passage 
of heat horizontally from one part of the atmosphere to another. 
Accordingly (9) is the form of equation appropriate to the 
problem, gravity, however, being evidently without percep- 
tible effect on the motion, so that the last principal term on. 
the right hand may be omitted. Thus 
^=<7-l)P^+7« 2 V%<- . . (17) 
To represent roughly the heating-effect of the sun, it is 
assumed that, during the day. heat is being gained per unit 
mass of air at a rate proportional to the sine of the sun's 
altitude ; while, day and night, heat is being lost per unit 
mass at a rate proportional to the fourth power of the absolute 
temperature of the air. As the fluctuations above and below 
the mean temperature in any latitude are supposed to he- 
small, the rate of loss referred to may be treated as constant. 
Taking, for simplicity, the time of an equinox, let = co- 
latitude, co — longitude. 2^ 
Then sin ( altitude of sun) = sin (Jd— co) sin 0. —7- being a 
solar day, and the origin of time being sunrise at the meridian 
Our assumptions are thus expressed by 
^?= A sin (kt-co) sin 0-BT 4 . 
d when 0<H-co<7T ; 
" " when 7r<Ll — co< 2tt. 
(18) 
To our order of approximation, \dH = for any whole day, 
which leads to 
BT* = £iH| (19) 
